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© 2018

Cubic Fields with Geometry

Benefits

  • Provides an up-to-date compendium of results

  • Helps the reader to envision what is explained in the text

  • Introduces the reader to several tools and disciplines which are applicable in the study of cubic fields

Book

Part of the CMS Books in Mathematics book series (CMSBM)

Table of contents

  1. Front Matter
    Pages i-xix
  2. Samuel A. Hambleton, Hugh C. Williams
    Pages 1-61
  3. Samuel A. Hambleton, Hugh C. Williams
    Pages 63-116
  4. Samuel A. Hambleton, Hugh C. Williams
    Pages 117-172
  5. Samuel A. Hambleton, Hugh C. Williams
    Pages 205-245
  6. Samuel A. Hambleton, Hugh C. Williams
    Pages 247-276
  7. Samuel A. Hambleton, Hugh C. Williams
    Pages 277-321
  8. Samuel A. Hambleton, Hugh C. Williams
    Pages 323-425
  9. Samuel A. Hambleton, Hugh C. Williams
    Pages 427-464
  10. Back Matter
    Pages 465-493

About this book

Introduction

The objective of this book is to provide tools for solving problems which involve cubic number fields. Many such problems can be considered geometrically; both in terms of the geometry of numbers and geometry of the associated cubic Diophantine equations that are similar in many ways to the Pell equation. With over 50 geometric diagrams, this book includes illustrations of many of these topics.  The book may be thought of as a companion reference for those students of algebraic number theory who wish to find more examples, a collection of recent research results on cubic fields, an easy-to-understand source for learning about Voronoi’s unit algorithm and several classical results which are still relevant to the field, and a book which helps bridge a gap in understanding connections between algebraic geometry and number theory.

The exposition includes numerous discussions on calculating with cubic fields including simple continued fractions of cubic irrational numbers, arithmetic using integer matrices, ideal class group computations, lattices over cubic fields, construction of cubic fields with a given discriminant, the search for elements of norm 1 of a cubic field with rational parametrization, and Voronoi's algorithm for finding a system of fundamental units. Throughout, the discussions are framed in terms of a binary cubic form that may be used to describe a given cubic field. This unifies the chapters of this book despite the diversity of their number theoretic topics. 

Keywords

binary cubic forms cubic fields Voronoi's algorithm geometry of numbers continued fractions fundamental units norm equation cubic Pell equation parametrization simultaneous approximation arithmetic matrices linear algebraic groups cubic Thue equations

Authors and affiliations

  1. 1.School of Mathematics and PhysicsThe University of QueenslandSt. Lucia, BrisbaneAustralia
  2. 2.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

Bibliographic information

  • Book Title Cubic Fields with Geometry
  • Authors Samuel A. Hambleton
    Hugh C. Williams
  • Series Title CMS Books in Mathematics
  • Series Abbreviated Title CMS Books in Mathematics
  • DOI https://doi.org/10.1007/978-3-030-01404-9
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-030-01402-5
  • eBook ISBN 978-3-030-01404-9
  • Series ISSN 1613-5237
  • Series E-ISSN 2197-4152
  • Edition Number 1
  • Number of Pages XIX, 493
  • Number of Illustrations 26 b/w illustrations, 27 illustrations in colour
  • Topics Algebraic Geometry
    Number Theory
    Algorithms
  • Buy this book on publisher's site
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